Publication at Faculty of Mathematics and Physics |
2005
Abstract
Let $\Omega$ be a convex planar domain and $f_\circ \in {\mathcal F}={\mathcal F}(\Omega , \Omega^\prime)$ where $\mathcal F$ is a class of $W^{1,1}$ homeomorphisms of finite distortion. We show that the minimization problem \begin{equation} \min_{f\in {\mathcal F}} \int_\Omega {\mathbb K}\, (x,f)\, dx , \; \; \; f=f_\circ \mbox{ on } \partial \Omega \end{equation} has a unique solution and that the extremal map is a ${\mathscr C}^\infty$-diffeomorphism whose inverse is harmonic in $\Omega^\prime$.